Question: Simplify; express your answer in exponential form. Assume $x\neq 0, t\neq 0$. $\dfrac{{(xt^{-1})^{5}}}{{(x^{-2}t^{-2})^{-5}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(xt^{-1})^{5} = (x)^{5}(t^{-1})^{5}}$ On the left, we have ${x}$ to the exponent ${5}$ . Now ${1 \times 5 = 5}$ , so ${(x)^{5} = x^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(xt^{-1})^{5}}}{{(x^{-2}t^{-2})^{-5}}} = \dfrac{{x^{5}t^{-5}}}{{x^{10}t^{10}}}$ Break up the equation by variable and simplify. $\dfrac{{x^{5}t^{-5}}}{{x^{10}t^{10}}} = \dfrac{{x^{5}}}{{x^{10}}} \cdot \dfrac{{t^{-5}}}{{t^{10}}} = x^{{5} - {10}} \cdot t^{{-5} - {10}} = x^{-5}t^{-15}$